Effective Floquet Hamiltonians for dipolar and quadrupolar coupled N-spin systems in solid state nuclear magnetic resonance under magic angle spinning

Spin dynamics under magic angle spinning has been studied using different theoretical approaches and also by extensive numerical simulation programs. In this article we present a general theoretical approach that leads to analytic forms for effective Hamiltonians for an N-spin dipolar and quadrupolar coupled system under magic angle spinning (MAS) conditions, using a combination of Floquet theory and van Vleck (contact) transformation. The analytic forms presented are shown to be useful for the study of MAS spin dynamics in solids with the help of a number of simulations in two, three, and four coupled, spin-1/2 systems as well as spins in which quadrupolar interactions are also present.     

Spin-locking of half-integer quadrupolar nuclei in nuclear magnetic resonance of solids: creation and evolution of coherences

Spin-locking of half-integer quadrupolar nuclei, such as 23Na (I=3/2) and 27Al (I=5/2), is of renewed interest owing to the development of variants of the multiple-quantum and satellite-transition magic angle spinning (MAS) nuclear magnetic resonance experiments that either utilize spin-locking directly or offer the possibility that spin-locked states may arise. However, the large magnitude and, under MAS, the time dependence of the quadrupolar interaction often result in complex spin-locking phenomena that are not widely understood. Here we show that, following the application of a spin-locking pulse, a variety of coherence transfer processes occur on a time scale of approximately 1/omegaQ before the spin system settles down into a spin-locked state which may itself be time dependent if MAS is performed. We show theoretically for both spin I=3/2 and 5/2 nuclei that the spin-locked state created by this initial rapid dephasing typically consists of a variety of single- and multiple-quantum coherences and nonequilibrium population states and we discuss the subsequent evolution of these under MAS. In contrast to previous work, we consider spin-locking using a wide range of radio frequency field strengths, i.e., a range that covers both the "strong-field" (omega1 > omegaQPAS and "weak-field" (omega1 < omegaQPAS limits. Single- and multiple-quantum filtered spin-locking experiments on NaNO2, NaNO3, and Al(acac)3, under both static and MAS conditions, are used to illustrate and confirm the results of the theoretical discussion.     

Theory of heteronuclear decoupling in solid-state nuclear magnetic resonance using multipole-multimode Floquet theory

A formal theory for heteronuclear decoupling in solid-state magic angle spinning (MAS) nuclear magnetic resonance experiments is presented as a first application of multipole-multimode Floquet theory. The method permits a straightforward construction of the multispin basis and describes the spin dynamics via effective Floquet Hamiltonians obtained using the van Vleck transformation method in the Floquet-Liouville space. As a test case, we consider a model three-spin system (I2S) under asynchronous time modulations (both MAS and rf irradiation) and derive effective Hamiltonians for describing the spin dynamics in the Floquet-Liouville space during heteronuclear decoupling. Furthermore, we describe and evaluate the origin of cross terms between the various anisotropic interactions and illustrate their exact contributions to the spin dynamics. The theory presented herein should be applicable to the design and understanding of pulse sequences for heteronuclear and homonuclear recoupling and decoupling.     

Satellite transitions acquired in real time by magic angle spinning (STARTMAS): "ultrafast" high-resolution MAS NMR spectroscopy of spin I=3/2 nuclei

The satellite transitions acquired in real time by magic angle spinning (STARTMAS) NMR experiment combines a train of pulses with sample rotation at the magic angle to refocus the first- and second-order quadrupolar broadening of spin I=3/2 nuclei in a series of echoes, while allowing the isotropic chemical and quadrupolar shifts to evolve. The result is real-time isotropic NMR spectra at high spinning rates using conventional MAS equipment. In this paper we describe in detail how STARTMAS data can be acquired and processed with ease on commercial equipment. We also discuss the advantages and limitations of the approach and illustrate the discussion with numerical simulations and experimental data from four different powdered solids.     

Nitrogen-14 NMR spectroscopy using residual dipolar splittings in solids

It is shown that nuclear magnetic resonance (NMR) spectra of nitrogen-14 (spin I = 1) can be obtained by indirect detection in powders spinning at the magic angle (MAS). The method relies on the transfer of coherence from a neighboring nucleus with S = 1/2, such as carbon-13, to single- or double-quantum transitions of nitrogen-14 nuclei. The transfer of coherence occurs through second-order quadrupole-dipole cross terms, also known as residual dipolar splittings. The two-dimensional NMR spectra reveal powder patterns determined by the second-order quadrupolar interactions of nitrogen-14. Analysis of the spectra yields the quadrupolar coupling constant, CQ, and asymmetry parameter, etaQ, of nitrogen-14. These parameters can be related to the structure of nitrogen-containing solids.     

Nylon 6 structure in solid state as studied by high-resolution 13C-NMR spectroscopy

High resolution ¹³C-NMR spectra of nylon 6 samples crystallized under various conditions and of a drawn sample were measured at room temperature by the cross polarization/magic angle spinning (CP/MAS) and pulse saturation transfer/magic angle spinning techniques. Additionally, ¹³C-NMR spectra of the drawn sample were measured at temperatures from 20 to 100°C by the CP/MAS technique and at 20 and 100°C by the low-power decoupling/magic angle spinning technique. The nylon 6 structure in the solid-state is discussed on the basis of these results. The solid-state ¹³C chemical shift data are used for reference in a study of conformation in solution.     

Characterization of binding affinities in a chromatographic system by suspended state HR/MAS NMR spectroscopy

In the current work a racemate of (R)‐ and (S)‐benzylmandelate was separated with a stereoselective polysaccharide‐based chiral stationary phase by HPLC. To elucidate the occurring chiral molecular recognition processes in the heterogeneous system used, NMR spectroscopy was chosen under high resolution/magic angle spinning (HR/MAS) NMR conditions in the suspended state. Therefore, and as a proof of concept, a combination of several NMR methods such as spin–lattice relaxation time (T₁) measurements (T₁), the saturation transfer difference, and the 2D experiment of the transferred nuclear overhauser enhancement spectroscopy technique were applied. With HR/MAS NMR it is feasible to combine NMR and chromatography to achieve further insights into the separation process. Copyright © 2009 John Wiley & Sons, Ltd.     

Unexpected effects of third-order cross-terms in heteronuclear spin systems under simultaneous radio-frequency irradiation and magic-angle spinning NMR

We recently noted [R. K. Harris, P. Hodgkinson, V. Zorin, J.-N. Dumez, B. Elena, L. Emsley, E. Salager, and R. Stein, Magn. Reson. Chem. 48, S103 (2010)] anomalous shifts in apparent (1)H chemical shifts in experiments using (1)H homonuclear decoupling sequences to acquire high-resolution (1)H NMR spectra for organic solids under magic-angle spinning (MAS). Analogous effects were also observed in numerical simulations of model (13)C,(1)H spin systems under homonuclear decoupling and involving large (13)C,(1)H dipolar couplings. While the heteronuclear coupling is generally assumed to be efficiently suppressed by sample spinning at the magic angle, we show that under conditions typically used in solid-state NMR, there is a significant third-order cross-term from this coupling under the conditions of simultaneous MAS and homonuclear decoupling for spins directly bonded to (1)H. This term, which is of the order of 100 Hz under typical conditions, explains the anomalous behaviour observed on both (1)H and (13)C spins, including the fast dephasing observed in (13)C{(1)H} heteronuclear spin-echo experiments under (1)H homonuclear decoupling. Strategies for minimising the impact of this effect are also discussed.     

Centerband‐only analysis of rotor‐unsynchronized spin echo for measurement of lipid 31P chemical shift anisotropy

Structural diversity and molecular flexibility of phospholipids are essential for biological membranes to play key roles in numerous cellular processes. Uncovering the behavior of individual lipids in membrane dynamics is crucial for understanding the molecular mechanisms underlying biological functions of cell membranes. In this paper, we introduce a simple method to investigate dynamics of lipid molecules in multi‐component systems by measuring the ³¹P chemical shift anisotropy (CSA) under magic angle spinning (MAS) conditions. For achieving both signal separation and CSA determination, we utilized a centerband‐only analysis of rotor‐unsynchronized spin echo (COARSE). This analysis is based on the curve fitting of periodic modulation of centerband intensity along the interpulse delay time in rotor‐unsynchronized spin‐echo experiments. The utility of COARSE was examined by using phospholipid vesicles, a three‐component lipid raft model system, and archaeal purple membranes. We found that the apparent advantages of this method are high resolution and high sensitivity given by the moderate MAS speed and the one‐dimensional acquisition with short spin‐echo delays. COARSE provides an alternative method for CSA measurement that is effective in the investigation of lipid polymorphologies. Copyright © 2015 John Wiley & Sons, Ltd.     

Solid-state 1H and 27Al NMR studies of amorphous aluminum hydroxides

Two kinds of amorphous aluminum hydroxides, a sample precipitated from admixing AlCl3 and NaOH aqueous solutions and the commercial product, were measured by 27Al and 1H solid-state NMR spectroscopy. Pentahedral and tetrahedral coordinations, as well as octahedral coordination of oxygen atoms for aluminum, are observed in 27Al magic angle spinning (MAS) spectra of both amorphous samples. In contrast, octahedral coordination is only observed in gibbsite, bayerite, and boehmite. According to 1H MAS-NMR spectra under conditions of high spinning rate (35 kHz) and high field (14.09 T), free waters and OH groups coupled with aluminum for amorphous samples are observed at approximately 5 and approximately 4.5 ppm, respectively, the latter peak being broader. This is consistent with the differential spectra between spin echo and transfer of populations in double resonance. We conclude that the subunits of AlO4, AlO5, and AlO6 in amorphous aluminum hydroxides are bound through hydrogen bonds with a wide distribution of bonding strength.     

Fiber spinning from the nematic melt. VI. Flow instabilities in the 30:70 copolyester of hydroxybenzoic acid and 2-hydroxy-6-naphthoic acid

The rheological behavior and fiber spinning are investigated for the Celanese liquid crystal copolyester 30 mol% p-hydroxybenzoic acid and 70 mol% 2-hydroxy-6-naphthoic acid (designated as 30HBA/70HNA) with inherent viscosity 7.8 dL/g. Shear thinning viscosity, and yield stress are observed at low shear stress, which probably results from the existence of crystallites in the melt. The crystal-nematic melting point of the copolymer, as measured by differential scanning calorimetry, is around 309°C. Extrudates are collected at four different temperatures ranging from 315 to 345°C. Melt fracture and die swell are observed above 335°C at low shear stress. A wide-angle x-ray diffraction (WAXS) study of an annealed sample indicates that the abnormal phenomenon may be due to crystallites arising from blocky units of HNA. Fiber spinning is performed at high shear rate at 325 and 335°C. Flow is stable under these conditions. The spin draw ratio is the ratio of take-up velocity to the velocity of extrudate existing from the capillary. The initial modulus reaches a maximum at a fairly low spin draw ratio. Instron and wide-angle x-ray (WAXS) studies show that the mechanical properties and orientation are poor for the fiber spun near the crystal-nematic melting point. Also, thermal history is found to affect the rheological behavior. Heat treatment offibers, particularly those which are well oriented, brings an improvement of mechanical properties.     

Nuclear Overhauser effect (NOE) enhancement of 11B NMR spectra of borane adducts in the solid state

A strong 11B {1H} nuclear Overhauser effect (NOE) enhancement can be observed in solid-state 11B NMR spectra of borane adducts, yielding fractional enhancements, fI{S} = (I - I0)/I0, of the magic angle spinning (MAS) NMR signal of up to 155%. This is an interesting and unusual observation as 11B (spin I = 3/2) is a quadrupolar nucleus and the corresponding NOE is completely absent in solution. More generally, it shows that the NOE may have a wider role to play in solid-state NMR studies of dynamics than has been envisaged hitherto.     

Proton spin diffusion in polyethylene as a function of magic-angle spinning rate. A phenomenological approach

Starting from the phenomenological Bloembergen-Purcell-Pound equation a relation between magic-angle spinning (MAS) rate and spin diffusion is derived. The resulting model equation was fitted to observed spin diffusion versus MAS rate data obtained at 298 K on an high-density polyethylene sample, revealing a reduction in the effective spin diffusivity by (65 + 5)% when increasing the MAS rate from 2 to 12 kHz. The same model equation enabled the rigid-lattice diffusivity to be estimated and was found to be only slightly higher, by approximately 10%, compared to the spin diffusivity observed at the lowest MAS rate applied (2 kHz). Moreover, the model equation predicts a reduction in the effective spin diffusivity by more than 90% when increasing the MAS rate to more than 30 kHz.     

Mechanism studies of the conversion of 13C-labeled n-butane on zeolite H-ZSM-5 by using 13C magic angle spinning NMR spectroscopy and GC-MS analysis

By using 13C MAS NMR spectroscopy (MAS = magic angle spinning), the conversion of selectively 13C-labeled n-butane on zeolite H-ZSM-5 at 430-470 K has been demonstrated to proceed through two pathways: 1) scrambling of the selective 13C-label in the n-butane molecule, and 2) oligomerization-cracking and conjunct polymerization. The latter processes (2) produce isobutane and propane simultaneously with alkyl-substituted cyclopentenyl cations and condensed aromatic compounds. In situ 13C MAS NMR and complementary ex situ GC-MS data provided evidence for a monomolecular mechanism of the 13C-label scrambling, whereas both isobutane and propane are formed through intermolecular pathways. According to 13C MAS NMR kinetic measurements, both pathways proceed with nearly the same activation energies (E(a) = 75 kJ mol(-1) for the scrambling and 71 kJ mol(-1) for isobutane and propane formation). This can be rationalized by considering the intermolecular hydride transfer between a primarily initiated carbenium ion and n-butane as being the rate-determining stage of the n-butane conversion on zeolite H-ZSM-5.     

Spin diffusion driven by R-symmetry sequences: applications to homonuclear correlation spectroscopy in MAS NMR of biological and organic solids

We present a family of homonuclear (13)C-(13)C magic angle spinning spin diffusion experiments, based on R2(n)(v) (n = 1 and 2, v = 1 and 2) symmetry sequences. These experiments are well suited for (13)C-(13)C correlation spectroscopy in biological and organic systems and are especially advantageous at very fast MAS conditions, where conventional PDSD and DARR experiments fail. At very fast MAS frequencies the R2(1)(1), R2(2)(1), and R2(2)(2) sequences result in excellent quality correlation spectra both in model compounds and in proteins. Under these conditions, individual R2(n)(v) display different polarization transfer efficiency dependencies on isotropic chemical shift differences: R2(2)(1) recouples efficiently both small and large chemical shift differences (in proteins these correspond to aliphatic-to-aliphatic and carbonyl-to-aliphatic correlations, respectively), while R2(1)(1) and R2(2)(2) exhibit the maximum recoupling efficiency for the aliphatic-to-aliphatic or carbonyl-to-aliphatic correlations, respectively. At moderate MAS frequencies (10-20 kHz), all R2(n)(v) sequences introduced in this work display similar transfer efficiencies, and their performance is very similar to that of PDSD and DARR. Polarization transfer dynamics and chemical shift dependencies of these R2-driven spin diffusion (RDSD) schemes are experimentally evaluated and investigated by numerical simulations for [U-(13)C,(15)N]-alanine and the [U-(13)C,(15)N] N-formyl-Met-Leu-Phe (MLF) tripeptide. Further applications of this approach are illustrated for several proteins: spherical assemblies of HIV-1 U-(13)C,(15)N CA protein, U-(13)C,(15)N-enriched dynein light chain DLC8, and sparsely (13)C/uniformly (15)N enriched CAP-Gly domain of dynactin. Due to the excellent performance and ease of implementation, the presented R2(n)(v) symmetry sequences are expected to be of wide applicability in studies of proteins and protein assemblies as well as other organic solids by MAS NMR spectroscopy.     

Theory of stochastic dipolar recoupling in solid-state nuclear magnetic resonance

Dipolar recoupling techniques in solid-state nuclear magnetic resonance (NMR) consist of radio frequency (rf) pulse sequences applied in synchrony with magic-angle spinning (MAS) that create nonzero average magnetic dipole-dipole couplings under MAS. Stochastic dipolar recoupling (SDR) is a variant in which randomly chosen rf carrier frequency offsets are introduced to cause random phase modulations of individual pairwise couplings in the dipolar spin Hamiltonian. Several aspects of SDR are investigated through analytical theory and numerical simulations: (1) An analytical expression for the evolution of nuclear spin polarization under SDR in a two-spin system is derived and verified through simulations, which show a continuous evolution from coherent, oscillatory polarization exchange to incoherent, exponential approach to equilibrium as the range of random carrier offsets (controlled by a parameter f(max)) increases; (2) in a many-spin system, polarization transfers under SDR are shown to be described accurately by a rate matrix in the limit of large f(max), with pairwise transfer rates that are proportional to the inverse sixth power of pairwise internuclear distances; (3) quantum mechanical interferences among noncommuting pairwise dipole-dipole couplings, which are a complicating factor in solid-state NMR studies of molecular structures by traditional dipolar recoupling methods, are shown to be absent from SDR data in the limit of large f(max), provided that coupled nuclei have distinct NMR chemical shifts.     

Dipolar recoupling of I = 1/2, S = 3/2 spin pairs with SEDOR for static and spinning samples

Spin-echo, double-resonance (SEDOR) dipolar recoupling experiments are illustrated on an I = 1/2, S = 3/2 spin system for static and spinning samples. An (15)N-(23)Na spin system is used to show that the simple pulse sequence is very effective in causing (15)N dipolar dephasing using either a (23)Na pi/2 recoupling pulse or a long radio-frequency (r.f.) recoupling pulse.     

Broadband inversion for MAS NMR with single-sideband-selective adiabatic pulses

We explain how and under which conditions it is possible to obtain an efficient inversion of an entire sideband family of several hundred kHz using low-power, sideband-selective adiabatic pulses, and we illustrate with some experimental results how this framework opens new avenues in solid-state NMR for manipulating spin systems with wide spinning-sideband (SSB) manifolds. This is achieved through the definition of the criteria of phase and amplitude modulation for designing an adiabatic inversion pulse for rotating solids. In turn, this is based on a framework for representing the Hamiltonian of the spin system in an NMR experiment under magic angle spinning (MAS). Following earlier ideas from Caravatti et al. [J. Magn. Reson. 55, 88 (1983)], the so-called "jolting frame" is used, which is the interaction frame of the anisotropic interaction giving rise to the SSB manifold. In the jolting frame, the shift modulation affecting the nuclear spin is removed, while the Hamiltonian corresponding to the RF field is frequency modulated and acquires a spinning-sideband pattern, specific for each crystallite orientation.